The main strategy for the proof of this lemma is similar to that for Lemma 2 in Cai and Liu [1]}. Without loss of generality, we can translate \(X\)  and assume that \({\rm E}{\mathbf {X}}= \mu  = 0\) .
First, we show the following property on \(\theta _{ij}\)
\(c\sigma _{ii}\sigma _{jj} \le \theta _{ij} \le C\sigma _{ii}\sigma _{jj}.\) 
